conquerthedragoncalculusfandomcom-20200214-history
Riemann Sums
'Riemann Sums' Riemann Sums is a method used to find the area under a curve on a graph. There are three diiferent types of Riemann Sums: #Left Riemann Sum #Right Riemann Sum #Midpoint Riemann Sum First, let us take a look at how to use Riemann Sums in a problem. Here is an example problem: Ex. Find the area under y = x^2 from 0 ---> 1 The First step is to divide the area into four different equal rectangles. (you can pick any number, but we will just use four) We are going to do Left Riemann Sum first. Next, since we know that the difference between each rectangle is 1/4, we plug in each rectangle into the formula: L(number of rectangles) = (the distance between rectangles) x (f (farthest left point) + f (second most left point) + f (thrid most) + f (fourth most)) (Notice that we do not use the point most right.) So now we simply plug in our values: L4= (1/4) x ( f(0) + f(1/4) + f(1/2) + f(3/4) ) Find f(0), f(1/4), f(1/2), f(3/4) by pluging it in to the original equation y = x^2 Then calculate: L4= 1/4 x (0 + 1/16 + 1/4 + 9/16) ---> L4 = 1/4 x 8.75 = .21875 Viola! and there you go! Left Riemann Sum = .21875 Next, we will find Right Riemann Sum. First, let us seperate the graph into four equal rectangles again: Now, we can plug the rectangles into the formula for Right Riemann Sum: R(number of rectangles) = (distance between rectangles) x ( f(farthest right point) + f(second farthest right point) + f(third most) + f(fourth most) ) (Notice this time that we did not use the left most point.) Now we simply plug in our values: R4= (1/4) x ( f(1) + f(3/4) + f(1/2) + f(1/4) ) Find f(1), f(3/4), f(1/2), f(1/4) by pluging it in to the original equation y = x^2 then calculate, R4= 1/4 x (1 + 9/16 + 1/4 + 1/16) ------> R4 = 1/4 x (1.875)= .46875 Boom! There you go again! Right Riemann Sum = .46875 Lastly, Lets find Midpoint Riemann Sum. We will once again seperate the graph into four equal rectangles: This time, notice that we divide each rectangle in half. Now, we will plug the points of the middle of the rectangles into the formula for Midpoint Riemann Sum: M(# of rectangles) = (distance between rectangles) x ( f(middle of 1/4 & 0 rectangle) + ( f(middle of 1/4 & 1/2) + f(middle of 1/2 & 3/4) + f(middle of 3/4 & 1) ) Now plug in the values: M4= 1/4 x ( f(1/8) + f(3/8) + f(5/8) + f(7/8) ) Find f(1/8), f(3/8), f(5/8), f(7/8) by pluging it in to the original equation y = x^2 M4= 1/4 x ( .015625 + .140625 + .390625 + .765625) -----> M4 = 1/4 x (1.3125)= .328125 AND KABOOOOOOOOOOM! THERE IT IS! Midpoint Riemann Sum = .328125 That is how you use Riemann Sums. Now it is time for a mini-quiz Here is some sample questions. #Find the area under the curve y = 1/2x from 0 to 2 2. find the area under the curve y = x^3 from 0 to 2